Seven Factorial

I know I have drawn this shape so often already, but the process of drawing it is so soothing.

And for that, I have drawn a kind of step-by-step guide how to draw that shape in the top of this drawing:

(from left to right: ) [sorry in advance if I make it sound more complicated than it actually is. If you want to draw it, I would advice you to focus more on these illustrations rather than on my gibberish-text.]

1. draw a 2-dimensional Cartesian plane - or, in other words: just draw a cross like depicted

1.1. mark 2 points on the y-axis/vertical line with same distance to the coordinate origin, then mark 2 points on the x-axis/horizontal line with the same distance to the coordinate origin. (The markings on the y-axis need to be farther away from the origin than the markings of the x-axis)

2. connect the 4 marked points like depicted above. This is a function plot of a tractrix. (it has two mirror symmetry axes. )

3. draw an ellipse and connect the two markings on the x-axis. This becomes a kind of "belt" for the pseudosphere (4th picture)

4. part the ellipse into whatever-amount-you-want of partings (like you would cut a cake) and slightly mark these.

5. now imagine you cut that shape horizontally on the outer surface. (In the 5th picture I depicted that with red-ish pen across the pseudosphere. ) -

6. then the cut shape needs to be "(shape) shifted". For that we use a set of marked points we did in step 4). Furtherly, we "cut" the ellipse open, and push one end of it to the top, and the other end to the bottom. (depicted in picture 6 )

7. Then we connect the rest to get that shape:

“I amsick will not to choir because i have a heache. i Hope its very and i am so sorry

love,

blue”

IT WAS ON THE DESKTOP. THIS IS WHAT I SENT.

I crocheted a series of hyperbolic planes and strung them up into a garland. Each module is one more round than the one before, and each round has twice the stitches as the round before.

I made eight iterations; the largest one took an entire skein of cotton yarn. I really liked how this shows the change in curvature as cross section of the plane expands.

Akatsuki's IR2 camera acquired this view of the night side of Venus. Infrared energy from hot, mid-altitude clouds shows up bright, while higher clouds that block the heat appear dark. The dark sawtooth at center appears to be a turbulent boundary. The planet's sunlit crescent is overexposed at upper right.

Most famous constants announce themselves immediately. For example, π appears wherever circles show up, e emerges from growth and calculus, and so on. But there's a stranger hiding deep within this infinite series:

ζ(3) = 1 + 1 / 2^3 + 1 / 3^3 + 1 / 4^3 + . . .

The number this sum converges to is called Apéry's constant, and despite looking innocent, it resisted proof for centuries. Mathematicians strongly suspected that it was irrational but nobody could prove it until 1978.

The Zeta Function Apéry's constant comes from one of the most important objects in mathematics: the Riemann zeta function.

For real numbers s > 1,

ζ(s) = ∑_{n=1}^∞ 1 / n^s

(the infinite series of 1 / 1 ^ s + 1 / 2 ^ s + 1 / 3 ^ s + . . . )

At first glance, this is just an infinite sum. But the zeta function secretly connects prime numbers, complex analysis, quantum physics, probability, cryptography, and the distribution of the primes themselves.

Some values are beautifully understood. For example,

ζ(2) = π^2 / 6

ζ(4) = π^4 / 90

In fact, every even positive integer produces a formula involving powers of π. But the odd inputs are another story.

Nobody knows a comparably elegant formula for

ζ(3), ζ(5), ζ(7), . . .

These numbers are mysterious, and ζ(3) became the first "battleground".

Roger Apéry's Bombshell

Numerically, Apéry's constant equals approximately

1.202056903159694...

The question sounds deceptively simple: Is this number rational? For over 200 years, nobody knew. That's remarkable because Euler had solved the analogous problem for ζ(2) in the 1700s.

In 1978, French Mathematician Roger Apéry announced that ζ(3) was irrational in a lecture. The announcement was met with criticism in part because Apéry was relatively unknown at the time. He also gave the lecture in French, made jokes throughout, and omitted several key explanations needed to follow the proof..

For example, there was an equation at the beginning of his lecture that no one knew but formed the core of his proof. When asked where this equation came from, Apéry is said to have answered "They grow in my garden," which was said to have caused many in the audience to stand up and leave the room.

However, someone in attendance had an electronic calculator (uncommon at the time) and with a short program, checked Apéry's equation and found it correct.

The equation in question is below, which was an unknown series representation of ζ(3) at the time:

With this expression, he was able to use a condition for irrationality that German mathematician Gustav Lejeune Dirichlet had derived in the 19th century. It states that a number χ is irrational if there are an infinite number of integers p and q, so that the following inequality is satisfied:

Here, c and δ denote constant values. Although the formula looks complicated, it basically means that χ can be approximated by fractions, but there is no fractional number that corresponds to χ exactly. Apéry succeeded in deriving this inequality for ζ(3), and thus the number is irrational.

In simpler terms, Apéry's proof constructed two sequences of integers:

a_n and b_n

such that

a_n / b_n

approximate ζ(3) far too well for a rational number.

This is the key philosophical idea. If a number is rational, there are limits to how accurately fractions can approximate it without eventually becoming exact. Apéry built approximations that violated those limits.

The machinery involved strange recursive sequences and combinatorial identities that seemed to come out of nowhere.

Even now, many mathematicians describe the proof as "magical".

To honor his work, the value of ζ(3) now bears his name and is known as Apéry's constant. This doesn't answer all of the questions associated with the number, however. We are still looking for a clear numerical value for ζ(3) that can be expressed with known constants, much in the same way ζ(2) is.

But regardless, Apéry's constant feels like an accident. It emerges from a simple series, has no known simple closed form, and required centuries to understand even partially. And yet it keeps appearing across math and physics like a recurring character in a story nobody fully understands.